Single view geometry Camera model Single view geom

Single view geometry Camera model Single view geom.

Pinhole camera image Amsterdam: what do you see in this picture? lstraight line lsize lparallelism/angle lshape of planes ldepth Photo by Robert Kosara, robert@kosara. net http: //www. kosara. net/gallery/pinholeamsterda m/pic 01. html

Pinhole camera image Amsterdam üstraight line lsize lparallelism/angle lshape of planes ldepth Photo by Robert Kosara, robert@kosara. net http: //www. kosara. net/gallery/pinholeamsterda m/pic 01. html

Pinhole camera image Amsterdam üstraight line ´size lparallelism/angle lshape of planes ldepth Photo by Robert Kosara, robert@kosara. net http: //www. kosara. net/gallery/pinholeamsterda m/pic 01. html

Pinhole camera image Amsterdam üstraight line ´size ´parallelism/angle lshape of planes ldepth Photo by Robert Kosara, robert@kosara. net http: //www. kosara. net/gallery/pinholeamsterda m/pic 01. html

Pinhole camera image Amsterdam üstraight line ´size ´parallelism/angle ´shape lshape of planes ldepth Photo by Robert Kosara, robert@kosara. net http: //www. kosara. net/gallery/pinholeamsterda m/pic 01. html

Pinhole camera image Amsterdam üstraight line ´size ´parallelism/angle ´shape lshape ü of planes parallel to image ldepth Photo by Robert Kosara, robert@kosara. net http: //www. kosara. net/gallery/pinholeamsterda m/pic 01. html

Pinhole camera image Amsterdam: what do you see? üstraight line ´size ´parallelism/angle ´shape lshape ü of planes parallel to image l. Depth ? lstereo - We see spatial shapes rather than individual pixels lmotion lsize lstructure … - Knowledge: top-down vision belongs to human - Stereo &　Motion most successful in 3 D CV & application - You can see it but you don't know how…

透視投影 (Perspective Projection) Pinhole Virtual image plane Image plane 簡略されたモデル： z y y’ O x x’ Object

同次座標系 Homogenous Coordinates

カメラのパラメータ Camera Parameters u 画像座標系Image coordinates 　　　(ximage, yimage) u 画像中心Image center 　　　　(ox, oy) u カメラ座標系Camera coordinates 　　(xcamera, ycamera) u u u ワールド座標系Real world coordinates (X, Y, Z) 焦点距離Focal length　　　　　 f 画素の有効サイズEffective size of pixel in millimeter 　　　　　　　　　　　

カメラの内部パラメータＩＩ (x, y) u u レンズのひずみ Lens Distortions k 1 , k 2 Modeled as simple radial distortions • r 2 = xd 2+yd 2 • (xd , yd) distorted points • k 1 , k 2: distortion coefficients • A model with k 2 =0 is still accurate for a CCD sensor of 500 x 500 with ~5 pixels distortion on the outer boundary (xd, yd)

平行移動（Translation） u (tx, ty, tz) Translation vector

平行移動（Translation） u Inverse translation

回転（Rotation） u u Inverse rotation 回転行列は直交行列!!

回転行列とEuler角 u u u 回転行列: 3 x 3 • 直交行列Orthogonal : , • 9 elements => 3+3 constraints (orthogonal) => 2+2 constraints (unit vectors) => 3 DOF 自由度 (degrees of freedom) Y , はX, Y, Z軸周りの回転角 X O 注意： Z a Zw • 一回一つの角度しか回転できない • 順番と関係がある Xw b Yw g

回転行列とEuler角 u If angle is small, then cos =1 and sin = また　 * + = Y O X Z a Zw Xw b Yw g

正射影モデル y (x’, y’, z’) (x, y, z) z O x Projection from (x, y, z) to (x’, y’, z’): or